Vibration control method for flapping-wing micro air vehicles

ABSTRACT

The present invention provides a method for controlling the oscillation of flapping-wing air vehicle, which comprises the following steps: calculating the kinetic energy, potential energy and virtual work of the system using the flexible wing with the two-degree of freedom as the research object; establishing system dynamics model based on the Hamilton&#39;s principle; setting the boundary control rate according to said system dynamics model wherein said boundary control rate includes F(t) and M(t), said F(t) is the inputted boundary control force, and said M(t) is the inputted boundary torque; and controlling the flexible wings according to the system dynamics model in combination with the boundary control rate. The present invention establishes the system dynamics model based on the Hamilton&#39;s principle, set the boundary control rate according to said system dynamics model, sufficiently considers the situation of distributed disturbance occurring at the boundary and effectively prevents the flexible wings deformation caused by the external disturbances.

CROSS-REFERENCE TO RELATED APPLICATIONS

This application claims priority benefit to Chinese Patent ApplicationNo. 201610169573.6 filed Mar. 23, 2016. The content of theaforementioned application, including any intervening amendmentsthereto, is incorporated herein by reference.

FIELD OF THE INVENTION

This invention relates to the field of automatic control technology, andespecially to a method for controlling the oscillation of flapping-wingair vehicle.

BACKGROUND TO THE INVENTION

In recent years, along with the continuous increase of people to theunmanned aerial vehicle (UAV) technology, and the rapid development ofthe advanced manufacturing technology (AMT), the new material technologyand the new energy technology, the research on the micro flying machineshave been the technical hot points.

Due to the increased demands to the UAV in civil and military usages,the designers are striving to reduce the weight of the UAV while improvethe maneuverability of the system. Therefore, at present, the flexiblewing with lighter weight is usually adopted in the UAV design. Comparedwith the rigid wing, the flexible wing mainly has the advantages of goodflexibility, good cost/benefit, better agility, and excellentperformance, etc. However, the flexible wing is easy to oscillate, andthus causing the unexpected error.

SUMMARY OF THE INVENTION

The technical problem to be solved by this invention is to provide amethod for controlling the oscillation of flapping-wing air vehiclewhich is able to effectively prevent the problem of flexible wingdeformation caused by the external disturbances.

The method for controlling the oscillation of flapping-wing air vehiclecomprises the steps of:

Calculating the kinetic energy, potential energy, and virtual work ofthe system using the flexible wing with the two-degree of freedom as theresearch object;

Establishing the system dynamics model based on Hamilton's principle;

Setting up the boundary control rate according to said system dynamicsmodel wherein said boundary control rate includes F(t)and M(t), saidF(t) is the inputted boundary control force, and said M(t) is theinputted boundary torque; and

Controlling the oscillation of the flexible wing according to the systemdynamics model and by combining the boundary control rate.

The advantageous effects of the present invention are as following:

-   the above technical solution establishes the system dynamics model    based on the Hamilton's principle, set the boundary control rate    according to said system dynamics model, considers the situation of    distributed disturbance occurring at the boundary sufficiently, and    effectively prevents the flexible wings deformation caused by the    external disturbances, thus is able to control the flexible wing    accurately and stably.

BRIEF DESCRIPTION OF THE SEVERAL VIEWS OF THE DRAWING

The invention will be described, by way of example, with reference tothe accompanying drawings, in which:

FIG. 1. is the flow diagram of the method of this invention forcontrolling the oscillation of the flapping-wing air vehicle;

FIG. 2 is the figure of simulation of the bending displacement under theinterference of the method of this invention for controlling theoscillation of the flapping-wing air vehicle;

FIG. 3. is the simulation of the torsional displacement under theinterference of the method of this invention for controlling theoscillation of the flapping-wing air vehicle.

DETAILED DESCRIPTION OF PREFERRED EMBODIMENT

In order to make the technical problem to be solved by this invention,the technical solution and the advantages clear, the invention will bedescribed by way of example, with reference to the accompanyingdrawings, in which:

As shown in FIG. 1, the method for controlling the oscillation offlapping-wing air vehicle, in the example of this invention, whichcomprises the steps of:

Step 101: calculating the kinetic energy, potential energy, and virtualwork of the system using the flexible wing with the two-degree offreedom as the research object;

Step 102: establishing the system dynamics model based on Hamilton'sprinciple,

Step 103: setting up the boundary control rate according to said systemdynamics model wherein said boundary control rate includes F(t) andM(t), said F(t) is the inputted boundary control force, and said M(t) isthe inputted boundary torque; and

Step 104: controlling the oscillation of the flexible wing according tothe system dynamics model and by combining the boundary control rate.

The method for controlling the oscillation of flapping-wing air vehiclein the example of this invention establishes the system dynamics modelbased on the Hamilton's principle, set the boundary control rateaccording to said system dynamics model, considers the situation ofdistributed disturbance occurring at the boundary sufficiently, andprevents the flexible wings deformation caused by the externaldisturbances effectively, thus is able to control the flexible wingaccurately and stably.

Preferably, said calculating the kinetic energy, potential energy, andvirtual work of the system using the flexible wing as the researchobject comprises:

Expressing the kinetic energy of the system, E_(k)(t) as follows:

$\begin{matrix}{{{E_{k}(t)} = {{\frac{1}{2}m{\int_{0}^{L}{\left\lbrack {\overset{.}{y}\left( {x,t} \right)} \right\rbrack^{2}{dx}}}} + {\frac{1}{2}\ I_{p}{\int_{o}^{L}{\left\lbrack {\overset{.}{\theta}\left( {x,t} \right)} \right\rbrack^{2}{dx}}}}}}\ ,} & (1)\end{matrix}$

wherein the spatial variable of x is independent to the time variable oft, and m is the unitspan mass of the flexible wing; I_(p) is inertialpolar distance of the flexible wing; y(x, t) is the bending displacementat the position of x and at time of t in the x0y coordinate system; andθ(x, t) is the corresponding displacement of deflection angle;

The potential energy of E_(p)(t) is expressed as follows:

$\begin{matrix}{{{E_{p}(t)} = {{\frac{1}{2}{EI}_{b}{\int_{0}^{L}{\left\lbrack {y^{''}\left( {x,t} \right)} \right\rbrack^{2}\ {dx}}}} + {\frac{1}{2}{GJ}{\int_{0}^{L}{\left\lbrack {\theta^{\prime}\left( {x,t} \right)} \right\rbrack^{2}\ {dx}}}}}},} & (2)\end{matrix}$

wherein, EI_(b) denotes the flexural rigidity, GJ denotes the torsionalrigidity; and the virtual work of caused by the above two rigiditiesδW_(c)(t) is expressed as follows:

δW _(c)(t)=mx _(o) c∫ ₀ ^(L) ÿ(x, t)δθ(x, t)dx+mx _(o) c∫ ₀ ^(L){umlautover (θ)}(x, t)δy(x, t)dx  (3),

Wherein, x_(o)c denotes the distance from the mass center of wing to thebending center; and the virtual work, δW_(d)(t) done by the Kelvin-Voigtdamping force is expressed as follows:

δW _(d)(t)=−ηEI _(b)∫₀ ^(L) {dot over (y)}″(x, t)δy″(x, t)dx−ηGJ _(b)∫₀^(L){dot over (θ)}′(x, t)δθ′(x, t)dx  (4),

Wherein, η denotes the Kelvin-Voigtd damping coefficient.

The virtual work of done by the distributed distraction, δW_(r)(t) is asfollows:

δW _(r)(t)=∫₀ ^(L) [F _(b)(x, t)δy(x, t)−x _(a) cF _(b)(x, t)δθ(x,t)]dx  (5),

wherein x_(a)c denotes the distance from the aerodynamic center to thebending centre and; F_(b) is the unknown time varying distributeddistraction along the wings;

The virtual work done by the boundary control force to the system,δW_(a)(t) is expressed as follows:

δW _(a)(t)=F(t)δy(L, t)+M(t)δθ(L, t)  (6),

In the above formula, F(t) is the inputted boundary control force and;M(t) is the inputted boundary torque;

Consequently, the total virtual work is:

δW(t)=δ[W _(c)(t)+W _(d)(t)+W _(r)(t)+W _(a)(t)]  (7).

Preferably, said establishing the system dynamics model based on theHamilton's principle includes:

utilizing the Hamilton's smooth action principle of

-   ∫_(t) ₁ ^(t) ² δ[E_(k)(t)−E_(p)(t)+W(t)]dt=0

Here δ denotes the variation symbol, and the governing equation for thesystem dynamics model is deduced as:

mÿ(x, t)+EI _(b) y″″(x, t)−mx _(o) c{umlaut over (θ)}(x, t)+ηEI _(b){dot over (y)}″″(x, t)=F _(b)(x, t)  (8),

I _(p){umlaut over (θ)}(x, t)−GJθ″(x, t)−mx _(o) cÿ(x, t)−ηGJ{dot over(θ)}″(x, t)=−x _(a) cF _(b)(x, t)  (9),

And the boundary conditions for the system dynamics model are deducedas:

y(0, t)=y′(0, t)=y″(L, t)=θ(0, t)=0  (10),

EI _(b) y′″(L, t)+ηEI _(b) {dot over (y)}′″(L, t)=−F(t)  (11) and

GJθ′(L, t)+ηGJ{dot over (θ)}′(L, t)=M(t)  (12).

Preferably, said setting the boundary controller based on the systemdynamics model includes two controlling laws of

-   -   F(t) and M(t), wherein said F(t) is the inputted boundary        control force and M(t) is the inputted boundary torque, and        which includes:

Constructing the Lyapunov candidate function as follows:

Preferably, said setting the boundary controller based on the systemdynamics model includes two controlling laws of

V(t)=V ₁+Δ(t)  (13),

Wherein, V₁(t) and Δ(t) are respectively defined as:

$\begin{matrix}{\left. {{V_{1}(t)} = {{\frac{\beta}{2}m{\int_{0}^{L}{{\left\lbrack {\overset{.}{y}\left( {x,t} \right)} \right\rbrack \ }^{2}{dx}}}} + {\frac{\beta}{2}{EI}_{b}{\int_{0}^{L}{{\left\lbrack {y^{''}\left( {x,t} \right)} \right\rbrack \ }^{2}{dx}}}} + {\frac{\beta}{2}I_{p}{\int_{0}^{L}{{\left\lbrack {\overset{.}{\theta}\left( {x,t} \right)} \right\rbrack \ }^{2}{dx}}}} + {\frac{\beta}{2}{GJ}{\int_{0}^{L}{\theta^{\prime}\left( {x,t} \right)}}}}} \right\rbrack 2\ {dx}\mspace{14mu} {and}} & (14) \\{{\Delta (t)} = {{\alpha \; m{\int_{0}^{L}{{\overset{.}{y}\left( {x,t} \right)}{y\left( {x,t} \right)}{dx}}}} + {\alpha \; I_{p}{\int_{0}^{L}{{\overset{.}{\theta}\left( {x,t} \right)}{\theta \left( {x,t} \right)}{dx}}}} - {\alpha \; {mx}_{e}c{\int_{0}^{L}{\left\lbrack {{{\overset{.}{y}\left( {x,t} \right)}{\theta \left( {x,t} \right)}} + {{y\left( {x,t} \right)}{\overset{.}{\theta}\left( {x,t} \right)}}} \right\rbrack {dx}}}} - {\beta \; {mx}_{e}c{\int_{0}^{L}{{\overset{.}{y}\left( {x,t} \right)}{\overset{.}{\theta}\left( {x,t} \right)}{{dx}.}}}}}} & (15)\end{matrix}$

In the above two equations, both α and β are the smaller positive weightcoefficient;

-   -   the boundary control rate is set by means of making the Lyapunov        candidate function be positive definite, and derivative of        Lyapunov candidate function of to the time of t, {dot over        (V)}(t) is negative definite.

Preferably, said calculating the boundary control rate when the Lyapunovcandidate function is positive definite, and the derivative of Lyapunovcandidate function the time of t, {dot over (V)}(t) is negative definitecomprises:

defining a new function as follows:

κ(t)=∫₀ ^(L) {[{dot over (y)}(x, t)]²+[{dot over (θ)}(x, t)]² +[y″(x,t)]²+[θ′(x, t)]² }dx  (16),

Then V₁(t) has the upper bound and lower bound which are defined as

γ₂κ(t)≦V ₁(t)≦γ₁κ(t)  (17),

In the above formula,

${\gamma_{1} = {\frac{\beta}{2}{\max \left( {m,I_{p},{EI}_{b},{GJ}} \right)}}},{\gamma_{2} = {\frac{\beta}{2}{{\min \left( {m,I_{p},{EI}_{b},{GJ}} \right)}.}}}$

Further, Δ(t) is magnified as

$\begin{matrix}{{{{\Delta (t)}} \leq {{\left( {{\alpha \; m} + {\alpha \; {mx}_{e}c} + {\beta \; {mx}_{e}c}} \right){\int_{0}^{L}{{\left\lbrack {\overset{.}{y}\left( {x,t} \right)} \right\rbrack \ }^{2}{dx}}}} + {\left( {{\alpha \; I_{p}} + {\alpha \; {mx}_{e}c} + {\beta \; {mx}_{e}c}} \right){\int_{0}^{L}{{\left\lbrack {\overset{.}{\theta}\left( {x,t} \right)} \right\rbrack \ }^{2}{dx}}}} + {\left( {{\alpha \; m} + {\alpha \; {mx}_{e}c}} \right)L^{4}{\int_{0}^{L}{{\left\lbrack {y^{''}\left( {x,t} \right)} \right\rbrack \ }^{2}{dx}}}} + {\left( {{\alpha \; I_{p}} + {\alpha \; {mx}_{e}c}} \right)L^{2}{\int_{0}^{L}{{\left\lbrack {\theta^{\prime}\left( {x,t} \right)} \right\rbrack \ }^{2}{dx}}}}} \leq {\gamma_{3}{\kappa (t)}}},} & (18)\end{matrix}$

Wherein,

-   γ₃=max{αm+αmx_(o)c+βmx_(o)c, αI_(p)+αmx_(o)c+βmx_(o)c,    (αm+αmx_(o)c)L⁴, (αI_(p)+αmx_(o)c)L²}, if the positive number, β    satisfies

${\beta > \frac{2\gamma_{3}}{\min \left( {m,I_{p},{EI}_{b},{GJ}} \right)}},$

then

0≦λ₂κ(t)≦V(t)≦λ₃κ(t)  (19).

This means that the constructed Lyapunov function is positive definite,wherein

-   λ₁=γ₁+γ₃ and λ₂=γ₂−γ₃;

The derivative of V(t) to t is deduced as:

{dot over (V)}(t)={dot over (V)} ₁(t)+{dot over (Δ)}(t)  (20), and

{dot over (V)} ₁(t)=βm∫ ₀ ^(L) {dot over (y)}(x, t)ÿ(x, t)dx+βI _(p)∫₀^(L){dot over (θ)}(x, t){umlaut over (θ)}(x, t)dx +βGJ∫ ₀ ^(L)θ′(x,t){dot over (θ)}′(x, t)dx+βEI _(b)∫₀ ^(L) y″(x, t){dot over (y)}″(x,t)dx  (21)

By introducing the controlling equations (8) and (9) into the aboveformula, we obtain:

{dot over (V)} ₁(t)=A ₁ +A ₂ +A ₃ +A ₄ +A ₅ +A ₆  (22).

Wherein, A₁-A₆ are respectively expressed as follows:

A ₁ =−βEI _(b)∫₀ ^(L) {dot over (y)}(x, t)y″″(x, t)dx+βEI _(b)∫₀ ^(L)y″(x, t){dot over (y)}″(x, t)dx  (23),

A ₂ =−βηEI _(b)∫₀ ^(L) {dot over (y)}(x, t){dot over (y)}″″(x,t)dx  (24),

A ₃ =βmx _(o) c ∫ ₀ ^(L) [{dot over (y)}(x, t){umlaut over (θ)}(x,t)+ÿ(x, t){dot over (θ)}(x, t)]dx  (25),

A ₄=β∫₀ ^(L) {dot over (y)}(x, t)F _(b)(x, t)dx−βx _(o) c∫ ₀ ^(L){dotover (θ)}(x, t)F _(b)(x, t)dx  (26),

A ₅ =βGJ∫ ₀ ^(L){dot over (θ)}(x, t)θ″(x, t)dx+βGJ∫ ₀ ^(L)θ′(x, t){dotover (θ)}′(x, t)dx  (27), and

A ₆ =βηGJ∫ ₀ ^(L){dot over (θ)}(x, t){dot over (θ)}″(x, t)dx  (28),

By utilizing the integration by parts and the boundary condition of (10)(11) and (12), we obtain

$\begin{matrix}{\mspace{79mu} {{A_{1} = {{{- \beta}\; {EI}_{b}\; {\overset{.}{y}\left( {L,t} \right)}{y^{\prime\prime\prime}\left( {L,t} \right)}} = {\beta \; {\overset{.}{y}\left( {L,t} \right)}{F(t)}}}},}} & (29) \\{{A_{2} \leq {{{- {\beta\eta}}\; {\overset{.}{y}\left( {L,t} \right)}{\overset{.}{F}(t)}} - {\frac{\beta \; \eta \; {EI}_{b}}{2L^{4}}{\int_{0}^{L}{{\left\lbrack {\overset{.}{y}\left( {x,t} \right)} \right\rbrack \ }^{2}{dx}}}} - {\frac{\beta \; \eta \; {EI}_{b}}{2}{\int_{0}^{L}{{\left\lbrack {{\overset{.}{y}}^{''}\left( {x,t} \right)} \right\rbrack \ }^{2}{dx}}}}}}\mspace{20mu} {and}} & (20) \\{{A_{4} \leq {{\sigma_{1}\beta {\int_{0}^{L}{{\left\lbrack {\overset{.}{y}\left( {x,t} \right)} \right\rbrack \ }^{2}{dx}}}} + {\sigma_{2}\beta \; x_{a}c{\int_{0}^{L}{{\left\lbrack {\overset{.}{\theta}\left( {x,t} \right)} \right\rbrack \ }^{2}{dx}}}} + {\left( {\frac{\beta}{\sigma_{1}} + \frac{\beta \; x_{a}c}{\sigma_{2}}} \right){LF}_{b\mspace{14mu} \max}^{2}}}},} & (31)\end{matrix}$

wherein, σ₁ and σ₂ are the positive constants, F_(b max) is the maximumvalue of the distributed disturbance, F_(b)(x, t).

$\begin{matrix}{\mspace{79mu} {{A_{5} = {{\beta \; {GJ}\; {\overset{.}{\theta}\left( {L,t} \right)}{\theta^{\prime}\left( {L,t} \right)}} = {\beta \; {\overset{.}{\theta}\left( {L,t} \right)}{M(t)}}}},}} & (32) \\{A_{6} \leq {{{\beta\eta}\; {\overset{.}{\theta}\left( {L,t} \right)}{\overset{.}{M}(t)}} - {\frac{\beta \; \eta \; {GJ}}{2L^{2}}{\int_{0}^{L}{{\left\lbrack {\overset{.}{\theta}\left( {x,t} \right)} \right\rbrack \ }^{2}{dx}}}} - {\frac{\beta \; \eta \; {GJ}}{2}{\int_{0}^{L}{{\left\lbrack {{\overset{.}{\theta}}^{\prime}\left( {x,t} \right)} \right\rbrack \ }^{2}{dx}}}}}} & (33)\end{matrix}$

Based on the above A1˜A6, we acquire {dot over (V)}₁(t) as follows:

$\begin{matrix}{{{\overset{.}{V}}_{1}(t)} \leq {{{- \left( {\frac{\beta \; \eta \; {EI}_{b}}{2L^{4}} - {\sigma_{1}\beta}} \right)}{\int_{0}^{L}{\left\lbrack {\overset{.}{y}\left( {x,t} \right)} \right\rbrack^{2}\ {dx}}}} - {\left( {\frac{\beta \; \eta \; {GJ}}{2L^{2}} - {\sigma_{2}\beta \; x_{a}c}}\; \right){\int_{0}^{L}{{\left\lbrack {\overset{.}{\theta}\left( {x,t} \right)} \right\rbrack \ }^{2}{dx}}}} - {\frac{\beta \; \eta \; {EI}_{b}}{2}{\int_{0}^{L}{\left\lbrack {{\overset{.}{y}}^{''}\left( {x,t} \right)} \right\rbrack^{2}\ {dx}}}} - {\frac{\beta \; \eta \; {GJ}}{2}{\int_{0}^{L}{{\left\lbrack {{\overset{.}{\theta}}^{\prime}\left( {x,t} \right)} \right\rbrack \ }^{2}{dx}}}} + {\beta \; {mx}_{e}c\; {\int_{0}^{L}{\left\lbrack {{{\overset{.}{y}\left( {x,t} \right)}{\overset{¨}{\theta}\left( {x,t} \right)}} + {{\overset{¨}{y}\left( {x,t} \right)}{\overset{.}{\theta}\left( {x,t} \right)}}} \right\rbrack \ {dx}}}} - {\beta \; {{\overset{.}{y}\left( {L,t} \right)}\left\lbrack {{F(t)} + {\eta \; {\overset{.}{F}(t)}}} \right\rbrack}} + {\beta {{\overset{.}{\theta}\left( {L,t} \right)}\left\lbrack {{M(t)} + {\eta \; {\overset{.}{M}(t)}}} \right\rbrack}} + {\left( {\frac{\beta}{\sigma_{1}} + \frac{\beta \; x_{a}c}{\sigma_{2}}} \right){{LF}_{b\mspace{14mu} \max}^{2}.}}}} & (34)\end{matrix}$

Similarly, by means of calculating the derivation of Δ(t) to t isdeduced as

{dot over (Δ)}(t)=B ₁ +B ₂ +. . . B ₈  (35),

B ₁ =−αEI _(b)∫₀ ^(L) y(x, t)y″″(x, t)dx  (36),

B ₂ =−αηEI _(b)∫₀ ^(L) y(x, t){dot over (y)}″″(x, t)dx  (37),

B ₃ =αGJ∫ ₀ ^(L)θ(x, t)θ″(x, t)dx  (38),

B ₄ =αηGJ∫ ₀ ^(L)θ(x, t){dot over (θ)}″(x, t)dx  (39),

B ₅ =αm∫ ₀ ^(L) [{dot over (y)}(x, t)]³ dx+αI _(p)∫₀ ^(L)[{dot over(θ)}(x, t)]² dx  (40),

B ₆ =−βmx _(o) c∫ ₀ ^(L) [{dot over (y)}(x, t){umlaut over (θ)}(x,t)+ÿ(x, t){dot over (θ)}(x, t)]dx  (41),

B ₇=−2αmx _(o) c∫ ₀ ^(L) {dot over (y)}(x, t){dot over (θ)}(x,t)dx  (42) and

B ₈=α∫₀ ^(L) y(x, t)F _(b)(x, t)dx−αx _(o) c∫ ₀ ^(L)θ(x, t)F _(b)(x,t)dx  (43).

By means of introducing the boundary conditions into the above formulas,we obtain:

$\begin{matrix}{\mspace{79mu} {{B_{1} = {{{- \alpha}\; {y\left( {L,t} \right)}{F(t)}} - {\alpha \; {EI}_{b}{\int_{0}^{L}{\left\lbrack {y^{''}\left( {x,t} \right)} \right\rbrack^{2}\ {dx}}}}}},}} & (44) \\{{B_{2} \leq {{{- \alpha}\; \eta \; {y\left( {L,t} \right)}{\overset{.}{F}(t)}} + {\frac{\alpha \; \eta \; {EI}_{b}}{\sigma_{3}}{\int_{0}^{L}{\left\lbrack {y^{''}\left( {x,t} \right)} \right\rbrack^{2}\ {dx}}}} + {\sigma_{3}\alpha \; \eta \; {EI}_{b}{\int_{0}^{L}{\left\lbrack {{\overset{.}{y}}^{''}\left( {x,t} \right)} \right\rbrack^{2}\ {dx}}}}}},} & (45) \\{\mspace{79mu} {{B_{3} = {{\alpha \; {\theta \left( {L,t} \right)}{M(t)}} - {\alpha \; {GJ}{\int_{0}^{L}{\left\lbrack {\theta^{\prime}\left( {x,t} \right)} \right\rbrack^{2}\ {dx}}}}}},}} & (46) \\{{B_{4} \leq {{\alpha \; \eta \; {\theta \left( {L,t} \right)}{\overset{.}{M}(t)}} + {\frac{\alpha \; \eta \; {GJ}}{\sigma_{4}}{\int_{0}^{L}{\left\lbrack {\theta^{\prime}\left( {x,t} \right)} \right\rbrack^{2}\ {dx}}}} + {\sigma_{4}\alpha \; \eta \; {GJ}{\int_{0}^{L}{\left\lbrack {{\overset{.}{\theta}}^{\prime}\left( {x,t} \right)} \right\rbrack^{2}\ {dx}}}}}},} & (47) \\{{B_{7} \leq {{2\alpha \; {mx}_{e}c\; \sigma_{5}{\int_{0}^{L}{\left\lbrack {\overset{.}{y}\left( {x,t} \right)} \right\rbrack^{2}\ {dx}}}} + {\frac{2\alpha \; {mx}_{e}c}{\sigma_{5}}{\int_{0}^{L}{\left\lbrack {\overset{.}{\theta}\left( {x,t} \right)} \right\rbrack^{2}\ {dx}}}}}},\mspace{20mu} {and}} & (48) \\{{B_{8} \leq {{\sigma_{6}\alpha \; L^{4}{\int_{0}^{L}{\left\lbrack {y^{''}\left( {x,t} \right)} \right\rbrack^{2}\ {dx}}}} + {\sigma_{7}\alpha \; x_{a}{cL}^{2}{\int_{0}^{L}{\left\lbrack {\theta^{\prime}\left( {x,t} \right)} \right\rbrack^{2}\ {dx}}}} + {\left( {\frac{\alpha}{\sigma_{6}} + \frac{\alpha \; x_{a}c}{\sigma_{7}}} \right){LF}_{b\mspace{11mu} \max}^{2}}}},} & (49)\end{matrix}$

All the above σ₃-σ₇ are the positive constants;

Therefore, according to B₁-B₈, we obtain:

$\begin{matrix}{{\overset{.}{\Delta}(t)} \leq {{{- \left( {{\alpha \; {EI}_{b}} - \frac{\alpha \; \eta \; {EI}_{b}}{\sigma_{3}} - {\sigma_{6}\alpha \; L^{4}}} \right)}{\int_{0}^{L}{\left\lbrack {y^{''}\left( {x,t} \right)} \right\rbrack^{2}\ {dx}}}} - {\left( {{\alpha \; {GJ}} - \frac{\alpha \; \eta \; {GJ}}{\sigma_{4}} - {\sigma_{7}\alpha \; x_{a}c\; L^{2}}} \right){\int_{0}^{L}{{\left\lbrack {\theta^{\prime}\left( {x,t} \right)} \right\rbrack \ }^{2}{dx}}}} + {\left( {{\alpha \; m} + {2\alpha \; {mx}_{e}c\; \sigma_{5}}} \right){\int_{0}^{L}{{\left\lbrack {\overset{.}{y}\left( {x,t} \right)} \right\rbrack \ }^{2}{dx}}}} + {\left( {{\alpha \; I_{p}} + \frac{2\alpha \; {mx}_{e}c}{\sigma_{5}}}\; \right){\int_{0}^{L}{{\left\lbrack {\overset{.}{\theta}\left( {x,t} \right)} \right\rbrack \ }^{2}{dx}}}} + {\sigma_{3}\alpha \; \eta \; {EI}_{b}{\int_{0}^{L}{{\left\lbrack {{\overset{.}{y}}^{''}\left( {x,t} \right)} \right\rbrack \ }^{2}{dx}}}} + {\sigma_{4}\alpha \; \eta \; {GJ}{\int_{0}^{L}{{\left\lbrack {{\overset{.}{\theta}}^{\prime}\left( {x,t} \right)} \right\rbrack \ }^{2}{dx}}}} - {\beta \; {mx}_{e}c{\int_{0}^{L}{{\left\lbrack {{{\overset{.}{y}\left( {x,t} \right)}{\overset{¨}{\theta}\left( {x,t} \right)}} + {{\overset{¨}{y}\left( {x,t} \right)}{\overset{.}{\theta}\left( {x,t} \right)}}} \right\rbrack \ }^{2}{dx}}}} + {\alpha \; {{\theta \left( {L,t} \right)}\left\lbrack {{M(t)} + {\eta \; {\overset{.}{M}(t)}}} \right\rbrack}} - {\alpha \; {{y\left( {L,t} \right)}\left\lbrack {{F(t)} + {\eta \; {\overset{.}{F}(t)}}} \right\rbrack}} + {\left( {\frac{\alpha}{\sigma_{6}} + \frac{\alpha \; x_{a}c}{\sigma_{7}}} \right){LF}_{b\mspace{14mu} \max}^{2}}}} & (50)\end{matrix}$

Based on the formulas of (34) and (50), we obtain:

$\begin{matrix}{{\overset{.}{V}(t)} \leq {{- {\left\lbrack {{\alpha \; {y\left( {L,t} \right)}} + {\beta \; {\overset{.}{y}\left( {L,t} \right)}}} \right\rbrack \left\lbrack {{F(t)} + {\eta \; {\overset{.}{F}(t)}}} \right\rbrack}} + {\left\lbrack {{\alpha \; {\theta \left( {L,t} \right)}} + {\beta \; {\overset{.}{\theta}\left( {L,t} \right)}}} \right\rbrack {\quad{\left\lbrack {{M(t)} + {\eta \; {\overset{.}{M}(t)}}} \right\rbrack - {\left( {{\alpha \; {EI}_{b}} - \frac{\alpha \; \eta \; {EI}_{b}}{\sigma_{3}} - {\sigma_{6}\alpha \; L^{4}}} \right){\int_{0}^{L}{\left\lbrack {y^{''}\left( {x,t} \right)} \right\rbrack^{2}\ {dx}}}} - {\left( {{\alpha \; {GJ}} - \frac{\alpha \; \eta \; {GJ}}{\sigma_{4}} - {\sigma_{7}\alpha \; x_{a}c\; L^{2}}} \right){\int_{0}^{L}{{\left\lbrack {\theta^{\prime}\left( {x,t} \right)} \right\rbrack \ }^{2}{dx}}}} - {\left( {\frac{\beta \; \eta \; {EI}_{b}}{2L^{4}} - {\sigma_{1}\beta} - {\alpha \; m} - {2\alpha \; {mx}_{e}c\; \sigma_{5}}} \right){\int_{0}^{L}{{\left\lbrack {\overset{.}{y}\left( {x,t} \right)} \right\rbrack \ }^{2}{dx}}}} - {\left( {\frac{\beta \; \eta \; {GJ}}{2L^{2}} - {\sigma_{2}\beta \; x_{a}c\; \alpha \; I_{p}} - {\alpha \; I_{p}} - \frac{2\alpha \; {mx}_{e}c}{\sigma_{5}}}\; \right){\int_{0}^{L}{{\left\lbrack {\overset{.}{\theta}\left( {x,t} \right)} \right\rbrack \ }^{2}{dx}}}} - {\left( {\frac{\beta \; \eta \; {EI}_{b}}{2} - {\sigma_{3}\alpha \; \eta \; {EI}_{b}}} \right){\int_{0}^{L}{\left\lbrack {{\overset{.}{y}}^{''}\left( {x,t} \right)} \right\rbrack^{2}\ {dx}}}} - {\left( {\frac{\beta \; \eta \; {GJ}}{2} - {\sigma_{4}\alpha \; \eta \; {GJ}}} \right){\int_{0}^{L}{\left\lbrack {{\overset{.}{\theta}}^{\prime}\left( {x,t} \right)} \right\rbrack^{2}\ {dx}}}} + {\left( {\frac{\beta}{\sigma_{1}} + \frac{\beta \; x_{a}c}{\sigma_{2}} + \frac{\alpha}{\sigma_{6}} + \frac{\alpha \; x_{a}c}{\sigma_{7}}} \right){{LF}_{b\mspace{11mu} \max}^{2}.}}}}}}} & (51)\end{matrix}$

By setting U(t)=F(t)+η{dot over (F)}(t) and V(t)=M(t)+η{dot over (M)}(t)as the new control variables, and their control rates are designed asfollows:

U(t)=k ₁ [αy(L, t)+β{dot over (y)}(L, t)]  (52),

V(t)=−k ₂[αθ(L, t)+β{dot over (θ)}(L, t)]  (53),

Wherein k₁≧0,k₂≧0 are the controlled gains.

Preferably, only need to set

${\frac{{\beta\eta}\; {EI}_{B}}{2} - {\sigma_{3}\alpha \; \eta \; {EI}_{b}}} \geq {0\mspace{14mu} {and}}$${{\frac{{\beta\eta}\; {GJ}}{2} - {\sigma_{4}\alpha \; \eta \; {GJ}}} \geq 0},$

we further obtain:

{dot over (V)}(t)≦μ₁∫₀ ^(L) [{dot over (y)}(x, t)]² dx−μ ₂∫₀ ^(L)[{dotover (θ)}(x, t)² dx −μ ₃∫₀ ^(L) [y″(x, t)]² dx−μ ₄∫₀ ^(L)[θ′(x, t)]²dx+ε−λ ₃κ(t)ε  (54),

Wherein

$\begin{matrix}{{\mu_{1} = {{\frac{\beta \; \eta \; {EI}_{b}}{2L^{4}} - {\sigma_{1}\beta} - {\alpha \; m} - {2\alpha \; {mx}_{e}c\; \sigma_{5}}} > 0}},} & (55) \\{{\mu_{2} = {{\frac{\beta \; \eta \; {GJ}}{2L^{2}} - {\sigma_{2}\beta \; x_{a}c\; \alpha \; I_{p}} - {\alpha \; I_{p}} - \frac{2\alpha \; {mx}_{e}c}{\sigma_{5}}}\; > 0}},} & (56) \\{{\mu_{3} = {{{\alpha \; {EI}_{b}} - \frac{\alpha \; \eta \; {EI}_{b}}{\sigma_{3}} - {\sigma_{6}\alpha \; L^{4}}}\; > 0}},} & (57) \\{{\mu_{4} = {{{\alpha \; {GJ}} - \frac{\alpha \; \eta \; {GJ}}{\sigma_{4}} - {\sigma_{7}\alpha \; x_{a}{cL}^{2}}}\; > 0}},} & (58) \\{\lambda_{3} = {{\min \left( {\mu_{1},\mu_{2},\mu_{3},\mu_{4}} \right)} > {0\mspace{14mu} {and}}}} & (59) \\{ɛ = {\left( {\frac{\beta}{\sigma_{1}} + \frac{\beta \; x_{a}c}{\sigma_{2}} + \frac{\alpha}{\sigma_{6}} + \frac{\alpha \; x_{a}c}{\sigma_{7}}} \right){LF}_{b\mspace{11mu} \max}^{2}}} & (60)\end{matrix}$

According to formulas of (19) and (54), we obtain:

{dot over (V)}(t)≦−λV(t)+ε  (61),

wherein λ=λ₃/λ₁, the above formula shows that only by means of selectingthe parameters, we can guarantee that {dot over (V)}(t) is negativedefinite.

Preferably, by integrating the inequation of (61), we obtain:

$\begin{matrix}{{{V(t)} \leq {{\left( {{V(0)} - \frac{ɛ}{\lambda}} \right)e^{{- \lambda}\; t}} + \frac{ɛ}{\lambda}} \leq {{{V(0)}e^{{- \lambda}\; t}} + \frac{ɛ}{\lambda}}} \in {L_{\infty}.}} & (62)\end{matrix}$

This means that V(t) is bounded. Further,

$\begin{matrix}{{{\frac{1}{L^{3}}{y^{2}\left( {x,t} \right)}} \leq {\frac{1}{L^{2}}{\int_{0}^{L}{\left\lbrack {y^{\prime}\left( {x,t} \right)} \right\rbrack^{2}\ {dx}}}} \leq {\int_{0}^{L}{\left\lbrack {y^{''}\left( {x,t} \right)} \right\rbrack^{2}\ {dx}}} \leq {\kappa (t)} \leq {\frac{1}{\lambda_{2}}{V(t)}}} \in L_{\infty}} & (63) \\{\mspace{79mu} {and}} & \; \\{\mspace{79mu} {{{\frac{1}{L}{\theta^{2}\left( {x,t} \right)}} \leq {\frac{1}{L^{2}}{\int_{0}^{L}{\left\lbrack {\theta^{\prime}\left( {x,t} \right)} \right\rbrack^{2}\ {dx}}}} \leq {\kappa (t)} \leq {\frac{1}{\lambda_{2}}{V(t)}}} \in L_{\infty}}} & (64)\end{matrix}$

is established, and thus we obtain:

$\begin{matrix}{{{{y\left( {x,t} \right)}} \leq \sqrt{\frac{L^{3}}{\lambda^{2}}\left( {{{V(0)}e^{{- \lambda}\; t}} + \frac{ɛ}{\lambda}} \right)}}{and}} & (65) \\{{{\theta \left( {x,t} \right)}} \leq {\sqrt{\frac{L}{\lambda^{2}}\left( {{{V(0)}e^{{- \lambda}\; t}} + \frac{ɛ}{\lambda}} \right)}.}} & (66)\end{matrix}$

When t tends to infinity, we obtain:

$\begin{matrix}{{{{y\left( {x,t} \right)}} \leq \sqrt{\frac{L^{3}ɛ}{\lambda_{2}\lambda}}},{\forall{x \in {\left\lbrack {0,L} \right\rbrack \mspace{14mu} {and}}}}} & (67) \\{{{{\theta \left( {x,t} \right)}} \leq \sqrt{\frac{L\; ɛ}{\lambda_{2}\lambda}}},{\forall{x \in {\left\lbrack {0,L} \right\rbrack \;.}}}} & (68)\end{matrix}$

This means that the system state y(x, t) and θ(x, t) are uniform bound.

To sum up, based on the Liapunov direct method, we can know that, bymeans of utilizing the boundary controls (52) and (53) to the systemsdescribed by the control equations (8) and (9) and the boundarycondition (10), (11) and (12), we can realize that the closed-loopsystem possesses the uniform boundedness properties.

The examples of this invention focused on the method for controlling theoscillation of flapping-wing air vehicle. Below, we will perform thenumerical simulation based on the MTLAB platform to verify the effect ofthe controller proposed for the problem of flexible wing deformation. Bymeans of adopting the finite-difference approximation, we obtained theapproximate values of the quantity of state in the formulas (8) and (9).The systematic parameters are shown in the following table:

TABLE 1 Table of parameters of the flexible wing of the air vehicleParameter Value L Length of the wings 2 m m Mass of per unitspan 10 kg/mI_(p) Polar moment of inertia of 1.5 kgm the intersecting surface of thewings EI_(b) Flexural rigidity 0.12 Nm² GJ Torsion resisting stiffness0.2 Nm² x_(c)c Distance from the wing 0.05 m center of mass to the shearcentre x_(a)c Distance from the 0.05 m aerodynamic center to the shearcentre η Kelvin-Voigt damping 0.05 coefficient

The starting conditions for the simulation is

${{y\left( {x,0} \right)} = \frac{x}{L}},{{\theta \left( {x,0} \right)} = \frac{\pi \; x}{2L}},{{\overset{.}{y}\left( {x,0} \right)} = 0},{{\overset{.}{\theta}\left( {x,0} \right)} = 0}$

when the distributed disturbance is F_(b)(x, t)=[1+sin (πt)+3 cos(3πt)]x.

The simulation diagrams 2 and 3 showed demonstrated that the boundarycontrollers designed in this invention are able to prevent thedeformation of the inflexible wing effectively.

The oscillation control device in this invention for the flapping-wingair vehicle adopted the method special to the flapping-wing air vehicle,so that the characteristics of the oscillation control device for theflapping-wing air vehicle are the same as those of the method foroscillation control of the flapping-wing air vehicle and won't be givenunnecessary details

What is said above is the preferred embodiment of this invention. Itshould be pointed out that the skilled person in the field of thistechnology is also able to think out a number of improvements andmodifications without far away from the principle stated in thisinvention, and these improvements and modifications are also should alsobe considered as the scope of protection of this invention.

We claim:
 1. A method for controlling the oscillation of flapping-wingair vehicle, comprising the following steps: Calculating the kineticenergy, potential energy, and virtual work of the system using theflexible wing as the research object; Establishing the Hamilton'sprinciple based system dynamics model; Setting the boundary control rateaccording to said system dynamics model wherein said boundary controlrate includes F(t) and M(t), said F(t) is the inputted boundary controlforce, and M(t) is the inputted boundary torque; and Controlling theflexible wings according to the system dynamics model and combining theboundary control rate.
 2. The method for controlling the oscillation offlapping-wing air vehicle of claim 1, characterized in that saidcalculating the kinetic energy, potential energy, and virtual work ofthe system using the flexible wing as the research object comprises: thekinetic energy of the system, E_(k)(t) is expressed as follows:$\begin{matrix}{{{E_{k}(t)} = {{\frac{1}{2}m{\int_{0}^{L}{\left\lbrack {\overset{.}{y}\left( {x,t} \right)} \right\rbrack^{2}\ {dx}}}} + {\frac{1}{2}I_{p}{\int_{0}^{L}{\left\lbrack {\overset{.}{\theta}\left( {x,t} \right)} \right\rbrack^{2}\ {dx}}}}}},} & (1)\end{matrix}$ wherein the spatial variable of x is independent to thetime variable of t, and m is the unitspan mass of the flexible wing;I_(p) is inertial polar distance of the flexible wing; y(x, t) is thebending displacement at the position of x and at time of t in the x0ycoordinate system; and θ(x, t) is the corresponding displacement ofdeflection angle; Potential energy of E_(p)(t) is expressed as follows:$\begin{matrix}{{{E_{p}(t)} = {{\frac{1}{2}{EI}_{b}{\int_{0}^{L}{\left\lbrack {y^{''}\left( {x,t} \right)} \right\rbrack^{2}\ {dx}}}} + {\frac{1}{2}{GJ}{\int_{0}^{L}{\left\lbrack {\theta^{\prime}\left( {x,t} \right)} \right\rbrack^{2}\ {dx}}}}}},} & (2)\end{matrix}$ wherein, EI_(b) denotes the flexural rigidity, and GJdenotes the torsional rigidity; and the virtual work of δW_(c)(t) causedby the above two rigidities is expressed as follows:δW _(c)(t)=mx _(o) c∫ ₀ ^(L) ÿ(x, t)δθ(x, t)dx+mx _(o) c∫ ₀ ^(L){umlautover (θ)}(x, t)δy(x, t)dx  (3), wherein x_(o)c denotes the distance fromthe mass center of wing to the bending center; and the virtual work ofδW_(d)(t) provided by the Kelvin-Voigt damping force is expressed asfollows:δW _(d)(t)=−ηEI _(b)∫₀ ^(L) {dot over (y)}″(x, t)δy″(x, t)dx−ηGJ _(b)∫₀^(L){dot over (θ)}′(x, t)δθ′(x, t)dx  (4), wherein, η denotes theKelvin-Voigtd damping coefficient; the virtual work of δW_(r)(t) done bythe distributed distraction is expressed as follows:δW _(r)(t)=∫₀ ^(L) [F _(b)(x, t)δy(x, t)−x _(a) cF _(b)(x, t)δθ(x,t)]dx  (5), wherein x_(a)c denotes the distance from the aerodynamiccenter to the bending centre and F_(b) is the unknown time varyingdistributed distraction along the wings; the virtual work of δW_(a)(t)done by the boundary control force to the system is expressed asfollows:δW _(a)(t)=F(t)δy(L, t)+M(t)δθ(L, t)  (6), In the above formula, F(t) isthe inputted boundary control force and M(t) is the inputted boundarytorque; Consequently, the total virtual work is:δW(t)=δ[W _(c)(t)+W _(d)(t)+W _(r)(t)+W _(a)(t)]  (7).
 3. The method forcontrolling the oscillation of flapping-wing air vehicle of claim 1,characterized in that, said establishing the system dynamics model basedon the Hamilton's principle includes: utilizing the Hamilton's smoothaction principle of ∫_(t) ₁ ^(t) ² δ[E_(k)(t)−E_(p)(t)+W(t)]dt=0 Here δdenotes the variation symbol, and the governing equation for the systemdynamics model is deduced as:mÿ(x, t)+EI _(b) y″″(x, t)−mx _(o) c{umlaut over (θ)}(x, t)+ηEI _(b){dot over (y)}″″(x, t)=F _(b)(x, t)  (8)I _(p){umlaut over (θ)}(x, t)−GJθ″(x, t)−mx _(o) cÿ(x, t)−ηGJ{dot over(θ)}″(x, t)=−x _(a) cF _(b)(x, t)  (9) And the boundary conditions forthe system dynamics model are deduced as:y(0, t)=y′(0, t)=y″(L, t)=θ(0, t)=0  (10),EI _(b) y′″(L, t)+ηEI _(b) {dot over (y)}′″(L, t)=−F(t)  (11) andGJθ′(L, t)+ηGJ{dot over (θ)}′(L, t)=M(t)  (12).
 4. The method forcontrolling the oscillation of flapping-wing air vehicle of claim 3,characterized in that, said setting the boundary controller based on thesystem dynamics model includes two controlling laws of F(t) and M(t)wherein said F(t) is the inputted boundary control force and said M(t)is the inputted boundary torque, and includes: Constructing the Lyapunovcandidate function as follows:V(t)=V ₁+Δ(t)  (13) Wherein, V₁(t) and Δ(t) are respectively defined as:$\begin{matrix}{{{V_{1}(t)} = {{\frac{\beta}{2}m{\int_{0}^{L}{\left\lbrack {\overset{.}{y}\left( {x,t} \right)} \right\rbrack^{2}\ {dx}}}} + {\frac{\beta}{2}{EI}_{b}{\int_{0}^{L}{\left\lbrack {y^{''}\left( {x,t} \right)} \right\rbrack^{2}\ {dx}}}} + {\left. \quad{{\frac{\beta}{2}I_{p}{\int_{0}^{L}{\left\lbrack {\overset{.}{\theta}\left( {x,t} \right)} \right\rbrack^{2}\ {dx}}}} + {\frac{\beta}{2}{GJ}{\int_{0}^{L}{\theta^{\prime}\left( {x,t} \right)}}}} \right\rbrack^{2}\ {dx}}}},} & (14) \\{{{\Delta (t)} = {{\alpha \; m{\int_{0}^{L}{{\overset{.}{y}\left( {x,t} \right)}\ {y\left( {x,t} \right)}{dx}}}} + {\alpha \; I_{p}{\int_{0}^{L}{{\overset{.}{\theta}\left( {x,t} \right)}{\theta \left( {x,t} \right)}\ {dx}}}} - {\alpha \; {mx}_{e}c{\int_{0}^{L}{\left\lbrack {{{\overset{.}{y}\left( {x,t} \right)}\ {\theta \left( {x,t} \right)}} + {{y\left( {x,t} \right)}{\overset{.}{\theta}\left( {x,t} \right)}}} \right\rbrack {dx}}}} - {\beta \; {mx}_{e}c{\int_{0}^{L}{{\overset{.}{y}\left( {x,t} \right)}{\overset{.}{\theta}\left( {x,t} \right)}\ {dx}}}}}};} & (15)\end{matrix}$ In the above two equations, both α and β are the smallerpositive weight coefficient; the boundary control rate is set by meansof making the Lyapunov candidate function be positive definite, andmaking the derivative of Lyapunov candidate function of {dot over(V)}(t) to the time of t be negative definite.
 5. The method forcontrolling the oscillation of flapping-wing air vehicle of claim 4,characterized in that, said calculating the boundary control rate whenthe Lyapunov candidate function is positive definite, and the derivativeof Lyapunov candidate function of {dot over (V)}(t) to the time of t isnegative definite includes: defining a new function as follows:κ(t)=∫₀ ^(L) {[{dot over (y)}(x, t)]²+[{dot over (θ)}(x, t)]² +[y″(x,t)]²+[θ′(x, t)]² }dx  (16), Then V₁(t) has the upper bound and lowerbound which are defined asγ₂κ(t)≦V ₁(t)≦γ₁κ(t)  (17), In the above formula,${\gamma_{1} = {\frac{\beta}{2}{\max \left( {m,I_{p},{EI}_{b},{GJ}} \right)}}},{{\gamma_{2} = {\frac{\beta}{2}{\min \left( {m,I_{p},{EI}_{b},{GJ}} \right)}}};}$Further, Δ(t) is magnified as: $\begin{matrix}{{{{\Delta \; (t)}} \leq {{\left( {{\alpha \; m} + {\alpha \; {mx}_{e}c} + {\beta \; {mx}_{e}c}} \right){\int_{0}^{L}{\left\lbrack {\overset{.}{y}\left( {x,t} \right)} \right\rbrack^{2}\ {dx}}}} + {\left( {{\alpha \; I_{p}} + {\alpha \; {mx}_{e}c} + {\beta \; {mx}_{e}c}} \right){\int_{0}^{L}{\left\lbrack {\overset{.}{\theta}\left( {x,t} \right)} \right\rbrack^{2}\ {dx}}}} + {\left( {{\alpha \; m} + {\alpha \; {mx}_{e}c}} \right)L^{4}{\int_{0}^{L}{\left\lbrack {y^{''}\left( {x,t} \right)} \right\rbrack^{2}\ {dx}}}} + {\left( {{\alpha \; I_{p}} + {\alpha \; {mx}_{e}c}} \right)L^{2}{\int_{0}^{L}{\left\lbrack {\theta^{\prime}\left( {x,t} \right)} \right\rbrack^{2}\ {dx}}}}} \leq {y_{3}{\kappa (t)}}},} & (18)\end{matrix}$ wherein γ₃=max{αm+αmx_(o)c+βmx_(o)c,αI_(p)+αmx_(o)c+βmx_(o)c, (αm+αmx_(o)c)L⁴, (αI_(p)+αmx_(o)c)L²}, if thepositive number of β satisfies${\beta > \frac{2\gamma_{3}}{\min \left( {m,I_{p},{EI}_{b},{GJ}} \right)}},$then0≦λ₂κ(t)≦V(t)≦λ₃κ(t)  (19), which means that the constructed Lyapunovfunction is positive definite, wherein λ₁=γ₁+γ₃ and λ₂=γ₂−γ₃; bycalculating the derivative of V(t) to t, we obtain:{dot over (V)}(t)={dot over (V)} ₁(t)+{dot over (Δ)}(t)  (20),{dot over (V)} ₁(t)=βm∫ ₀ ^(L) {dot over (y)}(x, t)ÿ(x, t)dx+βI _(p)∫₀^(L){dot over (θ)}(x, t){umlaut over (θ)}(x, t)dx +βGJ∫ ₀ ^(L)θ′(x,t){dot over (θ)}′(x, t)dx+βEI _(b)∫₀ ^(L) y″(x, t){dot over (y)}″(x,t)dx  (21) by introducing the controlling equation (8) and (9) into theabove formula, we obtain:{dot over (V)} ₁(t)=A ₁ +A ₂ +A ₃ +A ₄ +A ₅ +A ₆  (22), Wherein, A₁˜A₆are respectively expressed as followsA ₁ =−βEI _(b)∫₀ ^(L) {dot over (y)}(x, t)y″″(x, t)dx+βEI _(b)∫₀ ^(L)y″(x, t){dot over (y)}″(x, t)dx  (23),A ₂ =−βηEI _(b)∫₀ ^(L) {dot over (y)}(x, t){dot over (y)}″″(x,t)dx  (24),A ₃ =βmx _(o) c ∫ ₀ ^(L) [{dot over (y)}(x, t){umlaut over (θ)}(x,t)+ÿ(x, t){dot over (θ)}(x, t)]dx  (25),A ₄=β∫₀ ^(L) {dot over (y)}(x, t)F _(b)(x, t)dx−βx _(o) c∫ ₀ ^(L){dotover (θ)}(x, t)F _(b)(x, t)dx  (26),A ₅ =βGJ∫ ₀ ^(L){dot over (θ)}(x, t)θ″(x, t)dx+βGJ∫ ₀ ^(L)θ′(x, t){dotover (θ)}′(x, t)dx  (27), andA ₆ =βηGJ∫ ₀ ^(L){dot over (θ)}(x, t){dot over (θ)}″(x, t)dx  (28), Byutilizing the integration by parts and the ba nary condition of (10),(11) and (12), we obtain $\begin{matrix}{\mspace{79mu} {{A_{1} = {{{- \beta}\; {EI}_{b}{\overset{.}{y}\left( {L,t} \right)}{y^{\prime\prime\prime}\left( {L,t} \right)}} = {{- \beta}\; {\overset{.}{y}\left( {L,t} \right)}{F(t)}}}},}} & (29) \\{{A_{2} \leq {{{- \beta}\; \eta \; {y\left( {L,t} \right)}{\overset{.}{F}(t)}} - {\frac{\beta \; \eta \; {EI}_{b}}{2L^{4}}{\int_{0}^{L}{\left\lbrack {\overset{.}{y}\left( {x,t} \right)} \right\rbrack^{2}{dx}}}} - {\frac{\beta \; \eta \; {EI}_{b}}{2}{\int_{0}^{L}{\left\lbrack {{\overset{.}{y}}^{''}\left( {x,t} \right)} \right\rbrack^{2}\ {dx}}}}}},{and}} & (30) \\{{A_{4} \leq {{\sigma_{1}\beta {\int_{0}^{L}{\left\lbrack {\overset{.}{y}\left( {x,t} \right)} \right\rbrack^{2}\ {dx}}}} + {\sigma_{2}\beta \; x_{a}c{\int_{0}^{L}{\left\lbrack {\overset{.}{\theta}\left( {x,t} \right)} \right\rbrack^{2}\ {dx}}}} + {\left( {\frac{\beta}{\sigma_{1}} + \frac{\beta \; x_{a}c}{\sigma_{2}}} \right){LF}_{b\mspace{11mu} \max}^{2}}}},} & (31)\end{matrix}$ wherein and σ₁ and σ₂ are the positive constant, F_(b max)is the maximum value of the distributed disturbance of F_(b)(x, t);$\begin{matrix}{\mspace{79mu} {A_{5} = {{\beta \; {BJ}\; {\overset{.}{\theta}\left( {L,t} \right)}{\theta^{\prime}\left( {L,t} \right)}} = {\beta \; {\overset{.}{\theta}\left( {L,t} \right)}{M(t)}}}}} & (32) \\{A_{6} \leq {{\beta \; \eta \; {\overset{.}{\theta}\left( {L,t} \right)}{\overset{.}{M}(t)}} - {\frac{\beta \; \eta \; {GJ}}{2L^{2}}{\int_{0}^{L}{\left\lbrack {\overset{.}{\theta}\left( {x,t} \right)} \right\rbrack^{2}\ {dx}}}} - {\frac{\beta \; \eta \; {GJ}}{2}{\int_{0}^{L}{\left\lbrack {{\overset{.}{\theta}}^{\prime}\left( {x,t} \right)} \right\rbrack^{2}\ {dx}}}}}} & (33)\end{matrix}$ Based on the above A1˜A6, we acquire {dot over (V)}₁(t) asfollows: $\begin{matrix}{{{\overset{.}{V}}_{1}(t)} \leq {{{- \left( {\frac{\beta \; \eta \; {EI}_{b}}{2L^{4}} - {\sigma_{1}\beta}} \right)}{\int_{0}^{L}{\left\lbrack {\overset{.}{y}\left( {x,t} \right)} \right\rbrack^{2}\ {dx}}}} - {\left( {\frac{\beta \; \eta \; {GJ}}{2L^{2}} - {\sigma_{2}\beta \; x_{a}c}} \right){\int_{0}^{L}{\left\lbrack {\overset{.}{\theta}\left( {x,t} \right)} \right\rbrack^{2}\ {dx}}}} - {\frac{\beta \; \eta \; {EI}_{b}}{2}{\int_{0}^{L}{\left\lbrack {{\overset{.}{y}}^{''}\left( {x,t} \right)} \right\rbrack^{2}\ {dx}}}} - {\frac{\beta \; \eta \; {GJ}}{2}{\int_{0}^{L}{\left\lbrack {{\overset{.}{\theta}}^{\prime}\left( {x,t} \right)} \right\rbrack^{2}\ {dx}}}} + {\beta \; {mx}_{e}c{\int_{0}^{L}{\left\lbrack {{{\overset{.}{y}\left( {x,t} \right)}{\overset{¨}{\theta}\left( {x,t} \right)}} + {{\overset{¨}{y}\left( {x,t} \right)}{\overset{.}{\theta}\left( {x,t} \right)}}} \right\rbrack \ {dx}}}} - {\beta \; {{\overset{.}{y}\left( {L,t} \right)}\left\lbrack {{F(t)} + {\eta \; {\overset{.}{F}(t)}}} \right\rbrack}} + {\beta \; {{\overset{.}{\theta}\left( {L,t} \right)}\left\lbrack {{M(t)} + {\eta \; {\overset{.}{M}(t)}}} \right\rbrack}} + {\left( {\frac{\beta}{\sigma_{1}} + \frac{\beta \; x_{a}c}{\sigma_{2}}} \right){LF}_{b\mspace{11mu} \max}^{2}}}} & (34)\end{matrix}$ similarly, by means of calculating the derivation of Δ(t)to t, we obtain:{dot over (Δ)}(t)=B ₁ +B ₂ +. . . B ₈  (35),B ₁ =−αEI _(b)∫₀ ^(L) y(x, t)y″″(x, t)dx  (36),B ₂ =−αηEI _(b)∫₀ ^(L) y(x, t){dot over (y)}″″(x, t)dx  (37),B ₃ =αGJ∫ ₀ ^(L)θ(x, t)θ″(x, t)dx  (38),B ₄ =αηGJ∫ ₀ ^(L)θ(x, t){dot over (θ)}″(x, t)dx  (39),B ₅ =αm∫ ₀ ^(L) [{dot over (y)}(x, t)]³ dx+αI _(p)∫₀ ^(L)[{dot over(θ)}(x, t)]² dx  (40),B ₆ =−βmx _(o) c∫ ₀ ^(L) [{dot over (y)}(x, t){umlaut over (θ)}(x,t)+ÿ(x, t){dot over (θ)}(x, t)]dx  (41),B ₇=−2αmx _(o) c∫ ₀ ^(L) {dot over (y)}(x, t){dot over (θ)}(x,t)dx  (42), andB ₈=α∫₀ ^(L) y(x, t)F _(b)(x, t)dx−αx _(o) c∫ ₀ ^(L)θ(x, t)F _(b)(x,t)dx  (43); By means of introducing the boundary conditions into theabove formulas, we obtain: $\begin{matrix}{\mspace{79mu} {{B_{1} = {{{- \alpha}\; {y\left( {L,t} \right)}{F(t)}} - {\alpha \; {EI}_{b}{\int_{0}^{L}{\left\lbrack {y^{''}\left( {x,t} \right)} \right\rbrack^{2}\ {dx}}}}}},}} & (44) \\{{B_{2} \leq {{{- \; \alpha}\; \eta \; {y\left( {L,t} \right)}{\overset{.}{F}(t)}} + {\frac{\alpha \; \eta \; {EI}_{b}}{\sigma_{3}}{\int_{0}^{L}{\left\lbrack {y^{''}\left( {x,t} \right)} \right\rbrack^{2}\ {dx}}}} + {\sigma_{3}\; \alpha \; \eta \; {EI}_{b}{\int_{0}^{L}{\left\lbrack {{\overset{.}{y}}^{''}\left( {x,t} \right)} \right\rbrack^{2}\ {dx}}}}}},} & (45) \\{\mspace{79mu} {{B_{3} = {{\alpha \; {\theta \left( {L,t} \right)}{M(t)}} - {\alpha \; {GJ}{\int_{0}^{L}{\left\lbrack {\theta^{\prime}\left( {x,t} \right)} \right\rbrack^{2}\ {dx}}}}}},}} & (46) \\{{B_{4} \leq \; {{\alpha \; \eta \; {\theta \left( {L,t} \right)}{\overset{.}{M}(t)}} + {\frac{\alpha \; \eta \; {GJ}}{\sigma_{4}}{\int_{0}^{L}{\left\lbrack {\theta^{\prime}\left( {x,t} \right)} \right\rbrack^{2}\ {dx}}}} + {\sigma_{4}\; \alpha \; \eta \; {GJ}{\int_{0}^{L}{\left\lbrack {{\overset{.}{\theta}}^{\prime}\left( {x,t} \right)} \right\rbrack^{2}\ {dx}}}}}},} & (47) \\{{B_{7} \leq \; {{2\alpha \; {mx}_{e}c\; \sigma_{5}{\int_{0}^{L}{\left\lbrack {\overset{.}{y}\left( {x,t} \right)} \right\rbrack^{2}\ {dx}}}} + {\frac{2\alpha \; {mx}_{e}c}{\sigma_{5}}{\int_{0}^{L}{\left\lbrack {\overset{.}{\theta}\left( {x,t} \right)} \right\rbrack^{2}\ {dx}}}}}},{and}} & (48) \\{{B_{8} \leq {{\sigma_{6}\; \alpha \; L^{4}{\int_{0}^{L}{\left\lbrack {y^{''}\left( {x,t} \right)} \right\rbrack^{2}\ {dx}}}} + {\sigma_{7}\; \alpha \; x_{a}\; {cL}^{2}{\int_{0}^{L}{\left\lbrack {\theta^{\prime}\left( {x,t} \right)} \right\rbrack^{2}\ {dx}}}} + {\left( {\frac{\alpha}{\sigma^{6}} + \frac{\alpha \; x_{a}c}{\sigma_{7}}} \right){LF}_{b\mspace{11mu} \max}^{2}}}},} & (49)\end{matrix}$ and All the above σ₃-σ₇ are the positive constant.Therefore, according to B₁-B₈, we obtain formula of (50):$\begin{matrix}{{{\overset{.}{\Delta}(t)} \leq {{{- \; \left( {{\alpha \; {EI}_{b}} - \frac{{\alpha\eta}\; {EI}_{b}}{\sigma_{3}} - {\sigma_{6}\alpha \; L^{4}}} \right)}{\int_{0}^{L}{\left\lbrack {y^{''}\left( {x,t} \right)} \right\rbrack^{2}\ {dx}}}} - {\left( {{\alpha \; {GJ}} - \frac{{\alpha\eta}\; {GJ}}{\sigma_{4}} - {\sigma_{3}\; \alpha \; x_{a}c\; L^{2}}} \right){\int_{0}^{L}{\left\lbrack {\theta^{\prime}\left( {x,t} \right)} \right\rbrack^{2}\ {dx}}}} + {\left( {{\alpha \; m} + {2\; \alpha \; {mx}_{e}c\; \sigma_{5}}} \right){\int_{0}^{L}{\left\lbrack {\overset{.}{y}\left( {x,t} \right)} \right\rbrack^{2}\ {dx}}}} + {\left( \; {{\alpha \; I_{p}} + \frac{2\alpha \; {mx}_{e}c}{\sigma_{5}}} \right){\int_{0}^{L}{\left\lbrack {\overset{.}{\theta}\left( {x,t} \right)} \right\rbrack^{2}\ {dx}}}} + {\sigma_{3}{\alpha\eta}\; {EI}_{b}{\int_{0}^{L}{\left\lbrack {{\overset{.}{y}}^{''}\left( {x,t} \right)} \right\rbrack^{2}\ {dx}}}} + {\sigma_{4}{\alpha\eta}\; {GJ}{\int_{0}^{L}{\left\lbrack {{\overset{.}{\theta}}^{\prime}\left( {x,t} \right)} \right\rbrack^{2}\ {dx}}}} - {\beta \; {mx}_{e}c{\int_{0}^{L}{\left\lbrack {{{\overset{.}{y}\left( {x,t} \right)}{\overset{¨}{\theta}\left( {x,t} \right)}} + {{\overset{¨}{y}\left( {x,t} \right)}{\overset{.}{\theta}\left( {x,t} \right)}}} \right\rbrack \ {dx}}}} + {\alpha \; {{\theta \left( {L,t} \right)}\left\lbrack {{M(t)} + {\eta \; {\overset{.}{M}(t)}}} \right\rbrack}} - {\alpha \; {{y\left( {L,t} \right)}\left\lbrack {{F(t)} + {\eta \; {\overset{.}{F}(t)}}} \right\rbrack}} + {\left( {\frac{\alpha}{\sigma_{6}} + \frac{\alpha \; x_{a}c}{\sigma_{7}}} \right){LF}_{b\mspace{11mu} \max}^{2}}}},} & (50)\end{matrix}$ Based on the formulas of (34) and (50), we can obtain:$\begin{matrix}{{\overset{.}{V}(t)} \leq {{- {\left\lbrack {{\alpha \; {y\left( {L,t} \right)}} + {\beta \; {\overset{.}{y}\left( {L,t} \right)}}} \right\rbrack \left\lbrack {{F(t)} + {\eta \; {\overset{.}{F}(t)}}} \right\rbrack}} + {\left\lbrack {{\alpha \; {\theta \left( {L,t} \right)}} + {\beta \overset{.}{\; \theta}\left( {L,t} \right)}} \right\rbrack {\quad{{\left\lbrack {{M(t)} + {\eta \; {\overset{.}{M}(t)}}} \right\rbrack - {\left( {{\alpha \; {EI}_{b}} - \frac{{\alpha\eta}\; {EI}_{b}}{\sigma_{3}} - {\sigma_{6}\alpha \; L^{4}}} \right){\int_{0}^{L}{\left\lbrack {y^{''}\left( {x,t} \right)} \right\rbrack^{2}\ {dx}}}} - {\left( {{\alpha \; {GJ}} - \frac{{\alpha\eta}\; {GJ}}{\sigma_{4}} - {\sigma_{7}\; \alpha \; x_{a}c\; L^{2}}} \right){\int_{0}^{L}{\left\lbrack {\theta^{\prime}\left( {x,t} \right)} \right\rbrack^{2}\ {dx}}}} - {\left( {\frac{{\beta\eta}\; {EI}_{b}}{2L^{4}} - {\sigma_{1}\beta} - {\alpha \; m} - {2\alpha \; {mx}_{e}c\; \sigma_{5}}} \right){\int_{0}^{L}{\left\lbrack {\overset{.}{y}\left( {x,t} \right)} \right\rbrack^{2}\ {dx}}}} - {\left( {\frac{{\beta\eta}\; {GJ}}{2L^{2}} - {\sigma_{2}\beta \; x_{a}c\; \alpha \; I_{p}} - {\alpha \; I_{p}} - \frac{2\alpha \; {mx}_{e}c}{\sigma_{5}}}\; \right){\int_{0}^{L}{\left\lbrack {\overset{.}{\theta}\left( {x,t} \right)} \right\rbrack^{2}\ {dx}}}} - {\left( {\frac{\beta \; \eta \; {EI}_{b}}{2} - {\sigma_{3}\alpha \; \eta \; {EI}_{b}}} \right){\int_{0}^{L}{{\left\lbrack {{\overset{.}{y}}^{''}\left( {x,t} \right)} \right\rbrack \ }^{2}{dx}}}} - {\left( {\frac{{\beta\eta}\; {GJ}}{2} - {\sigma_{4}{\alpha\eta}\; {GJ}}} \right){\int_{0}^{L}{\left\lbrack {{\overset{.}{\theta}}^{\prime}\left( {x,t} \right)} \right\rbrack^{2}\ {dx}}}} + {\left( {\frac{\beta}{\sigma_{1}} + \frac{\beta \; x_{a}c}{\sigma_{2}} + \frac{\alpha}{\sigma_{6}} + \frac{\alpha \; x_{a}c}{\sigma_{7}}} \right){LF}_{b\mspace{11mu} \max}^{2}}},}}}}} & (51)\end{matrix}$ By setting U(t)=F(t)+η{dot over (F)}(t) andV(t)=M(t)+η{dot over (M)}(t) as the new controling variable, and theircontrol rates are designed as follows:U(t)=k ₁ [αy(L, t)+β{dot over (y)}(L, t)]  (52),V(t)=−k ₂[αθ(L, t)+β{dot over (θ)}(L, t)]  (53), Wherein k₁≧0,k₂≧0 isthe control gain.